= Mixed Heegaard Floer invariant
{c}
{title2=$\Phi_{M,\mathfrak s}$}
For a closed smooth oriented four-manifold $M$ with $b_2^+(M)>1$, remove two balls to obtain $W:S^3\to S^3$, and choose an <Admissible cut for a Heegaard Floer mixed map>. Positivity and naturality show that the minus map of $W_1$ lands in $HF^-_{\mathrm{red}}$, while the plus map of $W_2$ descends from $HF^+_{\mathrm{red}}$. Thus
$$
F^{\mathrm{mix}}_{W,\mathfrak s}=F^+_{W_2,\mathfrak s|W_2}\,\delta^{-1}\,F^-_{W_1,\mathfrak s|W_1}:HF^-(S^3)\longrightarrow HF^+(S^3).
$$
The coefficient of the bottom plus generator in this map, applied to $U^r\Theta^-\otimes\zeta$, defines $\Phi_{M,\mathfrak s}(U^r\otimes\zeta)\in\mathbb Z$, up to the orientation sign, with $\zeta\in\Lambda^*(H_1(M)/\mathrm{torsion})$. It vanishes unless $2r+\deg\zeta=(c_1(\mathfrak s)^2-2\chi(M)-3\sigma(M))/4$. The construction is independent of the admissible cut by the cobordism composition and associativity laws. Two positive pieces explain the condition $b_2^+>1$.
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